EXERCISES, DIRAC GEOMETRY - GQT SCHOOL
Exercise 1. Let ω ∈ Ω2 (M ) be a non-degenerate 2-form. For f ∈ C ∞ (M ), let
Xf ∈ X(M ) be the vector field defined by the equation:
df = ιXf ω.
Consider also the operation:
{f, g} := Xf (g), f, g ∈ C ∞ (M ).
Prove that ω is closed: dω = 0, iff {·, ·} satisfies the Jacobi identity:
{f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0.
Exercise 2. Let η ∈ Ω2 (M ) be a closed 2-form of constant rank. Prove that
ker(η) ⊂ T M is an involutive distribution.
Exercise 3. Let π ∈ X(M ) be a Poisson structure of constant rank. Prove that
π(T ∗ M ) ⊂ T M is an involutive distribution.
Exercise 4. Recall that the Dorfman bracket is defined by:
[v + α, w + β] = [v, w] + Lv β − ιw dα.
Prove that it satisfies the following relations:
(a) The Leibniz type rule:
[a1 , f a2 ] = LpT (a1 ) (f )a2 + f [a1 , a2 ].
(b) It is skew-symmetric up to an exact form:
[a1 , a2 ] + [a2 , a1 ] = d(a1 , a2 ).
(c) It satisfies the Jacobi identity written in the following form:
[a1 , [a2 , a3 ]] = [[a1 , a2 ], a3 ] + [a2 , [a1 , a3 ]].
(d) It preserves the metric (·, ·) in the following sense:
LpT (a1 ) (a2 , a3 ) = ([a1 , a2 ], a3 ) + (a2 , [a1 , a3 ]),
for all a1 , a2 , a3 ∈ Γ(TM ) and all f ∈ C ∞ (M ), where we have denoted
pT : TM −→ T M, pT (v + α) = v.
Exercise 5. Let L ⊂ TM be a Lagrangian subbundle.
(a) Show that the following defines an element NL ∈ Γ(∧3 L∗ ):
NL (α, β, γ) := ([a, b], c)(p),
where α, β, γ ∈ Lp and a, b, c ∈ Γ(L) are any sections extending α, β and γ,
respectively.
(b) Prove that L ∈ Dir(M ) if and only if NL = 0.
Exercise 6. (a) If ω ∈ Ω2 (M ) is closed, prove that T M ω ∈ Dir(M ), where
T M ω := {v + ιv ω : v ∈ T M }.
Conversely, if L ∈ Dir(M ) satisfies L ∩ T ∗ M = 0, prove that L = T M ω for
some closed 2-form ω.
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EXERCISES, DIRAC GEOMETRY - GQT SCHOOL
(b) If π ∈ X2 (M ) is a Poisson structure, prove that T ∗ Mπ ∈ Dir(M ), where
T ∗ Mπ := {π(α) + α : α ∈ T ∗ M }.
Conversely, if L ∈ Dir(M ) satisfies L ∩ T M = 0, prove that L = T ∗ Mπ for
some Poisson structure π. (Hint: use the tensor NL ).
Exercise 7. Let Lagr(V) denote the set of Lagrangian subspace of V = V ⊕ V ∗ .
For a pair (W, ω), where W ⊂ V is a linear space and ω ∈ ∧2 W ∗ define
L(W, ω) := {v + α : v ∈ W, α|W = ιv ω} ⊂ V.
Prove that L(W, ω) ∈ Lagr(V), and that this map
(W, ω) 7→ L(W, ω)
is a 1-to-1 correspondence between such pairs (W, ω) and Lagr(V).
Exercise 8. Prove that Lagr(V) has two connected components:
Lagr(V)i := {L(W, ω) : dim(W ) ≡ i mod 2}, i = 0, 1.
Conclude that, if L ∈ Dir(M ), where M is a connected manifold, then either all
leaves of L are odd-dimensional, or all leaves of L are even-dimensional.
We denote Dir(M )0 the Dirac structure with even-dimensional leaves and by
Dir(M )1 the Dirac structures with odd-dimensional leaves.
Exercise 9. (a) If dim(M ) = 1 and M is connected, show that Dir(M ) = {T M, T ∗ M }.
(b) If dim(M ) = 2, think about Dir(M )0 and Dir(M )1 .
(c) If dim(M ) = 3 and L ∈ Dir(M )0 , and f ∈ C ∞ (M ), prove that
ef L := {v + ef (x)α : v + α ∈ Lx , x ∈ M } ∈ Dir(M ).
Exercise 10. Let Ψ : TM → TM be a vector bundle isomorphism covering the
identity. If Ψ preserves the metric and the Dorfman bracket, prove that there exists
a unique closed 2-form ω so that Ψ = exp(ω), i.e.
Ψ(v + α) = v + α + ιv α.
Exercise 11. For L1 , L2 ∈ Lagr(V) define:
L1 ? L2 := {v + α + β : v + α ∈ L1 , v + β ∈ L2 }.
(a) Prove that L1 ? L2 ∈ Lagr(V).
(b) Prove that
L(W1 , ω1 ) ? L(W2 , ω2 ) = L(W1 ∩ W2 , ω1 |W1 ∩W2 + ω2 |W1 ∩W2 ).
(c) Prove that ? restricts to a smooth map on
{(L1 , L2 ) : prV (L1 ) + prV (L2 ) = V }.
Exercise 12. (a) Let L1 , L2 ∈ Dir(M ). If L1 and −L2 are transverse:
L1 ⊕ (−L2 ) = TM,
prove that L1 ? L2 is a Poisson structure.
(b) Let π1 , π2 ∈ X2 (M ) be two Poisson structures so that π1 + π2 is an invertible
bivector field. Prove that
π1 ? π2 := π1 ◦ (π1 + π2 )−1 ◦ π2
is a Poisson bivector field.
Exercise 13. For L ∈ Lagr(V2 ) and a linear map A : V1 → V2 , define:
A∗ L := {v1 + A∗ (α2 ) : A(v1 ) + α2 ∈ L}.
EXERCISES, DIRAC GEOMETRY - GQT SCHOOL
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(a) Show that A∗ L ∈ Lagr(V1 ), and that
A∗ (L(W, ω)) = L(A−1 (W ), A∗ ω).
(b) Show that (A, L) 7→ A∗ L is not continuous, but its restriction to the following
set is in fact smooth:
{(A, L) : A(V1 ) + prV2 (L2 ) = V2 }.
Exercise 14. Show that:
A∗ (L1 ? L2 ) = A∗ (L1 ) ? A∗ (L2 ),
but, that in general
A∗ (L1 ? L2 ) 6= A∗ (L1 ) ? A∗ (L2 ).
Exercise 15. Let f : M → N be a smooth map. Prove that f ∗ (T ∗ N ) ∈ Dir(M ) if
and only if f has constant rank.
Exercise 16. Let f : M → N be a smooth map, and let L ∈ Dir(N ) so that f ∗ (L)
is a Poisson structure. Prove that f is an immersion.
Exercise 17. Let (M, ω) be a symplectic manifold, and let f : M → N be a
surjective submersion with connected fibers. Denote by D the symplectic orthogonal
to the fibers of f :
Dx := {v ∈ Tx M : ω(v, w) = 0 ∀ w ∈ ker(dx f )}.
Prove Libermann’s Theorem: There exists a Poisson structure π on N so that
f : (M, ω) → (N, π) is a Poisson map, if and only if D is an involutive distribution.